Analysis of numerical methods based on modal shapes for the dynamic response analysis of aircraft structures

The thesis compares different methods used to obtain structural dynamic responses: Mode Displacement, Modal Truncation Augmentation, and Mode Acceleration. There is limited existing literature on these methods, with only a few recent scientific articles providing further insights into their differences and capabilities. The thesis conducts detailed analyses of the mentioned methods and also compares the "Inertia Relief-based" method to the Modal Truncation Augmentation method. Both methods are implemented by default in MSC Nastran's RESVEC function, which calculates residual vectors to obtain improved structural dynamic responses. Chapter 1 discusses the first methods, while Chapter 2 focuses on their implementation, code validation using reference literature data, and application to the Euler-Bernoulli beam finite element model. This section also aims to numerically and analytically compare the methods using different defined performance coefficients. Analyses show that the methods exhibit different behaviours depending on the frequency content, spatial distribution of the applied loads, and the system's natural frequencies. Therefore, enhancement approaches are designed to optimize the solution at each applied load’s frequency. Finally, in Chapter 3, it is provided an analytical comparison of the methods already implemented in MSC Nastran. This is achieved through the definition of other performance coefficients, conceptually similar to those used in the previous chapter. The effectiveness of the methods is evaluated by their adherence to the ideal solution and their computational cost, the latter is directly related to the total number of solved 1-DOF differential equations, which is equal to the combined number of retained and residual modal shapes. The results obtained in Chapter 2 indicate possible reasons why MSC Nastran's algorithm progressively adopted the Modal Truncation method over the Mode Acceleration method, whereas results from Chapter 3 suggest on which occasions it is more convenient to use the residual vector methods adopted by MSC Nastran's RESVEC function. It is anticipated that the Modal Truncation method is more effective than the Inertia Relief-based method at higher numbers of retained modes, providing the best approximation of the ideal solution at a reasonable computational cost.